<p>Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(v(G), r(G), \alpha (G)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <i>c</i>(<i>G</i>) be respectively the order, the rank, the independence number and the cyclomatic number of a graph <i>G</i>. Li et al. (Linear Multiliear Algebra 67:2230–2245, 2019) characterized all graphs <i>G</i> with rank <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(r(G)=2v(G)-2\alpha (G)-2c(G).\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>v</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mn>2</mn> <mi>α</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mn>2</mn> <mi>c</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> In this paper, we characterize all graphs <i>G</i> with rank <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(r(G)=2v(G)-2\alpha (G)-2c(G)+1.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>v</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mn>2</mn> <mi>α</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mn>2</mn> <mi>c</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>